If they did, they didn't show it in their writing. When we find accounts of disease in both medical and 'laymen's' works, Greek and Roman authors present sickness as something that happens to you, and survival as something out of the patient's hands - usually some mixture of the physician's skill, the circumstances around you, the gods' favour and the direction of fate, depending on who exactly is writing. If the patient has any say in the matter at all, it's somewhat incidental.

'Convalesce as Comfortably as Possible': Tiro's Fevers

I like the illustrative case study, so let's dive in with one. In 53 BC, Tiro, who was the favourite slave of the Roman lawyer, politician and philosopher Cicero, came down with what was, if we believe Cicero, a fairly serious fever. Tiro was lucky in that his master particularly liked him, and sent him to recuperate at one of his villas near Cumae. There have survived a series of letters which Cicero wrote to him during his convalescence, where he was very keen to ensure that Tiro recovered as fully as possible. Here's a good one to start with, written on April 17:

*ego vero cupio te ad me venire, sed viam timeo. gravissime aegrotasti, inedia et purgationibus et vi ipsius morbi consumptus es; graves solent offensiones esse ex gravibus morbis si quae culpa commissa est.*^(2)

I would very much like for you to come to me, but I'm worried about the journey. You have been seriously ill, and wasted away by fasting, purgatives and the workings of the disease itself; when it comes to serious illnesses, slip-ups are often serious, if anything is done wrongly.

This is interesting for two reasons. Firstly, it tells us a lot about how Roman doctors would treat a disease. Our main surviving sources for ancient medicine are the physician Galen (a Greek writing in the Roman period) and the looser collection of texts known as the 'Hippocratic Corpus', written by various Greek medics mostly in the 5th and 4th centuries BCE. They often disagree on the fine details, but are pretty unanimous in the belief that disease comes from an imbalance of 'humors' - vital substances like blood, bile and phlegm - in the body. Therefore, the best way to treat a disease is to correct the balance - in Tiro's case, to limit what goes in and to forcibly expel some of what was considered to be the excess. So there's a degree of personal agency here, but not really personal will. Tiro can't wish himself to be better any more than he can wish for the bile to leave his body.

The other reason I find it interesting is that Cicero was a massive busybody, and this is one of the many letters where he's keen to tell Tiro exactly what to do. There's very little 'fight' or indeed energy to his advice - the best thing for Tiro to do is to rest. Cicero does use the phrase cura ut valeas ('see to it that you get better'), and this is invariably what he means - avoid work, travel or any sort of stress.

Tiro was a fairly sickly man, and in 50 BC, while serving with Cicero in his governorship of Cilicia in Asia, he became ill again. Once again, Cicero sent him away to recover, this time to Issus. Once again, he wrote to him with very similar advice:

*magnae nobis est sollicitudini valetudo tua. nam tametsi qui veniunt ἀκίνδυνα μὲν χρονιώτερα δὲ nuntiant, tamen in magna consolatione ingens inest sollicitudo, si diutius a nobis afuturus es ... at tamen quamquam videre te tota cogitatione cupio, tamen te penitus rogo ne te tam longae navigationi et viae per hiemem nisi bene firmum committas neve naviges nisi explorate. vix in ipsis tectis et oppidis frigus infirma valetudine vitatur, nedum in mari et via sit facile abesse ab iniuria temporis. 'ψῦχος δὲ λεπτῷ χρωτὶ πολεμιώτατον, inquit Euripides.*^(3)

I am really very worried about your health. For although the people who come to me from you say 'it's not dangerous, just long-lasting', still that great consolation contains the anxiety that you will be away from me still longer ... But although I wholeheartedly want to see you, still I earnestly ask you not to put yourself through a long journey in the winter, unless you are truly better - not to sail and not to travel. Even in houses and towns, it's difficult to avoid the cold, never mind to avoid being harmed by the weather at sea or on the road. 'Cold is the greatest enemy of tender skin', as Euripides said.

Similar advice - but here we also see the belief that climate played a major role in recovery. In both cases of Tiro's illness, Cicero sent him away from a busy city into somewhere quieter, and this was common practice - extremes of cold, heat and humidity were considered harmful to health, and you can see the same sort of balance-seeking thinking at work as in the theory of humorism. Once again, though, it's the exact opposite of the vigorous, fighting language you've described in your post. Indeed, another bit of advice from the later stages of Tiro's illness makes the point well:

*modo fac, id quod est humanitatis tuae, ne quid aliud cures hoc tempore nisi ut quam commodissime convalescas. non ignoro quantum ex desiderio labores...*^(4)

Just make sure - as fits an educated man like you - that you don't do anything at this time except what is going to help you convalesce as comfortably as possible. I know how little you like being away from work...